# conductivityTensor¶

Specify a tensor that has different conductivity parallel and perpendicular to a given vector field. The Tensor is specified as

given an input vector field $$B$$ that same field is used to generate a unit vector field $$b$$ that is then used to define the tensor. Conductivity parallel to the magnetic field is specified as $$K_{\parallel}$$ and perpendicular to the vector field as $$K_{\perp}$$ and then the difference in conductivities $$dK=K_{\parallel}-K_{\perp}$$. The 9 tensor components are given as

\notag \begin{align} \left( \begin{array}{ccc} K_{\perp}+dK\,b_{x}^{2} & K_{\parallel}b_{x}b_{y} & K_{\parallel}b_{x}b_{z} \\ K_{\parallel}\,b_{x}b_{y} & K_{\perp}+dK\,b_{y}^{2} & K_{\parallel}b_{y}b_{z} \\ K_{\parallel}\,b_{x}b_{z} & K_{\parallel}b_{y}b_{z} & K_{\perp}+dK\,b_{z}^{2} \\ \end{array} \right) \end{align}

## Parent Updater Data¶

in (string vector, required)

1st Variable

1. $$V_{x}$$ x vector component
2. $$V_{y}$$ y vector component
3. $$V_{z}$$ z vector component

2nd Variable

1. $$K_{\parallel}$$ parallel conductivity

3rd Variable

1. $$K_{\perp}$$ perpendicular conductivity
out (string vector, required)

The output variable is a length 9 vector containing the 9 components of the conductivity tensor

1st Variable

1. $$T_{xx}$$
2. $$T_{xy}$$
3. $$T_{xz}$$
4. $$T_{yx}$$
5. $$T_{yy}$$
6. $$T_{yz}$$
7. $$T_{zx}$$
8. $$T_{zy}$$
9. $$T_{zz}$$

## Example¶

 <Updater initConductivityTensor>
kind = equation2d
onGrid = domain

in = [B, kParallel, kPerpendicular]

out = [conductivityTensor]

<Equation a>
kind = conductivityTensor
</Equation>
</Updater>